A short survey on stable polynomials, orientations and matchings

Abstract

This is a short survey about the theory of stable polynomials and its applications. It gives self-contained proofs of two theorems of Schrijver. One of them asserts that for a d-regular bipartite graph G on 2n vertices, the number of perfect matchings, denoted by pm(G), satisfiesThe other theorem claims that for even d the number of Eulerian orientations of a d-regular graph G on n vertices, denoted by ε(G), satisfiesTo prove these theorems we use the theory of stable polynomials, and give a common generalization of the two theorems.